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Full text of "Treatise On Analysis Vol-Ii"

272 XIV INTEGRATION IN LOCALLY COMPACT GROUPS is integrable with respect to the product measure fa ® u2 ® * • * ® \in on Gn. It follows immediately from this definition and from (13.21.17) that the sequence G"»)i^i^« is convolvable if and only if the sequence (|^il)i^^w of absolute values of these measures is convolvable. Moreover, it is clear that •J-J H\ (xn) is then a positive linear form on JTR(G) and therefore a positive measure on G (13.3.1) ; and for all/e JfH(G) we have (14.5.1 ) I f • • • f by virtue of (13.16.5) and (13.21.17). It follows directly (13.1.1) that /H> f • • • \ f(x1x2 ' ' ' xn) d^i(#i) • • * d^nCO is also a (complex) measure on G. This measure is denoted by ut * ^2 * • - - * un and is called the convolution pro- duct or convolution of the sequence (u^ ^t^n- The formula (1 4.5.1 ) also shows that (14.5.2) I/*! * /*2 * - • • * 0J ^ K| * |/i2| * • • • * |/ij. For each function/e «5fc(G), the function (xl9 . . . , ^n)i-^/(jc1A:2 • • • xn) is continuous and therefore measurable with respect to every measure on Gn; hence, by virtue of (13.21 .10), the sequence (jii9 . . . , nn) is convolvable if and only if (14.5.3) f *d |^(1)| (x,(1)) \ 4- oo for some permutation a of {1, 2, ...,«}. An equivalent condition is that, for each compact subset K of G, the (closed) set A c G" of points (xl9 x2,..., xn) such that xtx2 • • * JCB e K is (^ ® ju2 ® • • • ® ^-integrable. If a sequence (/z, v) of two measures is convolvable, we say that ^ and v (in this order) are convolvable, or that \i is convolvable on the left with v, or that v is convolvable on the right with u. If ^ and v are convolvable and /z', v' are two measures such that |ju'| ^ |ju| and |v'| ^ |v|, then it is clear from (14.5.3) that n' and v' are convolvable. measures